If you have a contingency case the lawyers split any legal fee. If you are paying an hourly fee you generally have to pay the lawyer what you owe and the lawyer can hold your file until you pay - this is called a retaining lien. The new lawyer usually prepares a Consent To Change Attorney form.
The above answer, and any follow up comments or emails is for informational purposes only and not meant as legal advice.
To add to Mr.Rothstein's answer, if it is a contingent retainer, what the first lawyer gets depends on how far along the case is when he or she is discharged. If the discharge takes place before any action is filed the first lawyer may be entitled to little or nothing. If the case is ready for trial and especially if there has been a substantial settlement offer, the first lawyer may be entitled to most of the fee. Sometimes lawyers negotiate the spit, sometimes the court has to decide.
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If you are changing lawyers, the "new" lawyer will have you sign a "consent to change attorney" form (a/k/a "substitution of attorney") and will then take the steps necessary to get your file from the first attorney. As already noted, depending on whether the first lawyer had an hourly fee retainer as compared with a contingency fee retainer will determine if you have to pay the first attorney any fees before your new attorney can get the file. Your new attorney will explain all of this to you.
There are two types of liens that the outgoing lawyer has: a retaining lien and a charging lien.
The outgoing lawyer will have a retaining lien for the case expenses that he/she incurred while representing you. When you fire the lawyer, you must pay the case expenses before your lawyer will send your case file to your new lawyer.
The outgoing lawyer will have a charging lien if a lawsuit has been filed. A charging lien is the lawyer's lien for a portion of the legal fee. Typically, the division of the legal fee between the outgoing and incoming lawyers is determined at the end of your case. The client can demand that the outgoing lawyer give you an accounting of their time and services when you switch attorneys, so their legal fee will be determined before the new attorney takes over responsibility for your case.
Your lawyer will not send your case file to the new attorney until you've paid the outstanding case expenses and there is some agreement regarding his/her right to a legal fee at the conclusion of your case.
You must have the outgoing lawyer sign a "Substitution of Attorneys" form that will be filed with the court and then served on opposing counsel.